Linear Equations and Graphing

Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane.’

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See (Figure 1).

‘Quadrant’ has the root ‘quad,’ which means ‘four.’

In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is . The point has a special name. It is called the origin.

We use the coordinates to locate a point on the xy-plane. Let’s plot the point as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through . Then, locate 3 on the y-axis and sketch a horizontal line through . Now, find the point where these two lines meet—that is the point with coordinates .

Notice that the vertical line through and the horizontal line through are not part of the graph. We just used them to help us locate the point .

How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.

For the point in (Figure 2) in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?

Can you tell just by looking at the coordinates in which quadrant the point is located? In which quadrant is located?

What if one coordinate is zero as shown in (Figure 3)? Where is the point located? Where is the point located?                             

The point is on the y-axis and the point is on the x-axis.

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the ordered pair use the correct order, .

Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like . (Then, you checked the solution by substituting back into the equation.)
Here’s an example of an equation in one variable, and its one solution.

But equations can have more than one variable. Equations with two variables may be of the form . Equations of this form are called linear equations in two variables.

Notice the word line in linear. Here is an example of a linear equation in two variables, and .

The equation is also a linear equation. But it does not appear to be in the form . We can use the Addition Property of Equality and rewrite it in form.

Add to both sides.
Simplify.
Use the Commutative Property to put it in form.

By rewriting as , we can easily see that it is a linear equation in two variables because it is of the form . When an equation is in the form , we say it is in standard form.

Most people prefer to have , , and be integers and when writing a linear equation in standard form, although it is not strictly necessary.

Linear equations have infinitely many solutions. For every number that is substituted for there is a corresponding value. This pair of values is a solution to the linear equation and is represented by the ordered pair . When we substitute these values of and into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

 

Complete a Table of Solutions to a Linear Equation in Two Variables

In the examples above, we substituted the x– and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for and then solve the equation for . Or, pick a value for and then solve for .

We’ll start by looking at the solutions to the equation that we found in (Example 5). We can summarize this information in a table of solutions, as shown in (Table 1).

Table 1

0
1	4

To find a third solution, we’ll let and solve for .

                                                                 

The ordered pair is a solution to . We will add it to (Table 2).

Table 2

0
1	4
2	9

We can find more solutions to the equation by substituting in any value of or any value of and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for or . But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of and then solve for .

We have seen how using zero as one value of makes finding the value of easy. When an equation is in standard form, with both the and on the same side of the equation, it is usually easier to first find one solution when find a second solution when , and then find a third solution.

Glossary

Practice Makes Perfect

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

1.A B C D E	2. A B C D E
3. A B C D E	4. A B C D E

In the following exercises, plot each point in a rectangular coordinate system.

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

9.	10.
11.	12.

Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

13. A B C	14. A B C
15. A B C	16. A B C
17. A B C	18. A B C
19. A B C	20. A B C

Everyday Math

49. Weight of a baby. Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column. a) Plot the points on a coordinate plane. b) Why is only Quadrant I needed? Age Weight 0 7 (0, 7) 2 11 (2, 11) 4 15 (4, 15) 6 16 (6, 16) 8 19 (8, 19) 10 20 (10, 20) 12 21 (12, 21)

50. Weight of a child. Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column. a) Plot the points on a coordinate plane. b) Why is only Quadrant I needed? Height Weight 28 22 (28, 22) 31 27 (31, 27) 33 33 (33, 33) 37 35 (37, 35) 40 41 (40, 41) 42 45 (42, 45)

Answers

1.	3.
5.	7.
9. A:  B:  C:  D:	11. A:  B:  C:  D:
13. A, B	15. A, C
17. B, C	19. A, B
21. 0 2 0	23. 0 5 3 2 7
25. 0 1 3 2 6 3	25. 0 1 3 2 6 3
27. 0 2 1	29. 0 2 3 4 6 0
31. 0 10 2 5 0	33. Answers will vary.
35. Answers will vary.	37. Answers will vary.
39. Answers will vary.	41. Answers will vary.
43. Answers will vary.	45. Answers will vary.
47. Answers will vary.	49. a) b) Age and weight are only positive.
51. Answers will vary.	53. Answers will vary.

Attributions

This chapter has been adapted from “Use the Rectangular Coordinate System” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Recognize the Relationship Between the Solutions of an Equation and its Graph

In the previous section, we found several solutions to the equation . They are listed in the table below. So, the ordered pairs , , and are some solutions to the equation . We can plot these solutions in the rectangular coordinate system as shown in (Figure 1).

0	3
2	0
1

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation . See (Figure 2). Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.

Notice that the point whose coordinates are is on the line shown in (Figure 3). If you substitute and into the equation, you find that it is a solution to the equation.

So the point is a solution to the equation . (The phrase “the point whose coordinates are ” is often shortened to “the point .”)

So is not a solution to the equation . Therefore, the point is not on the line. See (Figure 2). This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation .

Graph Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just and no , or just without an ? How will we make a table of values to get the points to plot?

Let’s consider the equation . This equation has only one variable, . The equation says that is always equal to , so its value does not depend on . No matter what is, the value of is always .

So to make a table of values, write in for all the values. Then choose any values for . Since does not depend on , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See the table below.

	1
	2
	3

Plot the points from the table and connect them with a straight line. Notice in (Figure 8) that we have graphed a vertical line.

0	4
2	4
4	4

The graph is a horizontal line passing through the y-axis at 4. See (Figure 10).

The equation has both and . The value of depends on the value of . The y-coordinate changes according to the value of . The equation has only one variable. The value of is constant. The y-coordinate is always 4. It does not depend on the value of . See the table below.

0	0			0	4
1	4			1	4
2	8			2	4

Notice, in (Figure 12), the equation gives a slanted line, while gives a horizontal line.

Key Concepts

• Graph a Linear Equation by Plotting Points
  1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
  3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Glossary

Practice Makes Perfect

Recognize the Relationship Between the Solutions of an Equation and its Graph

In the following exercises, for each ordered pair, decide:

a) Is the ordered pair a solution to the equation? b) Is the point on the line?

1. a) b) c) d)	2. a) b) c) d)
3. a) b) c) d)	4. a) b) c) d)

In the following exercises, graph by plotting points.

5.	6.
7.	8.
9.	10.
11.	12.
13.	14.
15.	16.
17.	18.
19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.
33.	34.
35.	36.
37.	38.
39.	40.
41.	42.
43.	44.
45.	46.
47.	48.

In the following exercises, graph each equation.

49.	50.
51.	52.
53.	54.
55.	56.
57.	58.
59.	60.

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

61. and	62. and
63. and	64. and

Answers

1. a) yes; no b) no; no c) yes; yes d) yes; yes	3. a) yes; yes b) yes; yes c) yes; yes d) no; no
5.	7.
9.	11.
13.	15.
17.	19.
21.	23.
25.	27.
29.	31.
33.	35.
37.	39. *ANSWER GRAPH LOOKS OFF; ie. graph should have m=2/5, not (-2/5).
41.	43.
45.	47.
49.	51.
53.	55.
57.	59.
61.	63.
65.	67.
69.	71.
73.	75.
77.	79.
81. $722, $850, $978	83. Answers will vary.

Attributions

This chapter has been adapted from “Graph Linear Equations in Two Variables” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Identify the x– and y– Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x– axis and the y– axis. These points are called the intercepts of the line.

Let’s look at the graphs of the lines in (Figure 1).

Examples of graphs crossing the x-negative axis.

First, notice where each of these lines crosses the negative axis. See (Figure 1).

Figure	The line crosses the x– axis at:	Ordered pair of this point
Figure (a)	3
Figure (b)	4
Figure (c)	5
Figure (d)	0

Do you see a pattern?

For each row, the y– coordinate of the point where the line crosses the x– axis is zero. The point where the line crosses the x– axis has the form and is called the x– intercept of a line. The x– intercept occurs when is zero.

Now, let’s look at the points where these lines cross the y– axis. See the table below.

Figure	The line crosses the y-axis at:	Ordered pair for this point
Figure (a)	6
Figure (b)
Figure (c)
Figure (d)	0

What is the pattern here?

In each row, the x– coordinate of the point where the line crosses the y– axis is zero. The point where the line crosses the y– axis has the form and is called the y- intercept of the line. The y– intercept occurs when is zero.

Find the x– and y– Intercepts from an Equation of a Line

Recognizing that the x– intercept occurs when y is zero and that the y– intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x– intercept, let and solve for x. To find the y– intercept, let and solve for y.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x– and y– intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Key Concepts

• Find the x– and y– Intercepts from the Equation of a Line
  • Use the equation of the line to find the x– intercept of the line, let and solve for x.
  • Use the equation of the line to find the y– intercept of the line, let and solve for y.
• Graph a Linear Equation using the Intercepts
  1. Find the x– and y– intercepts of the line.
     Let and solve for x.
     Let and solve for y.
  2. Find a third solution to the equation.
  3. Plot the three points and then check that they line up.
  4. Draw the line.
• Strategy for Choosing the Most Convenient Method to Graph a Line:
  • Consider the form of the equation.
  • If it only has one variable, it is a vertical or horizontal line.
    is a vertical line passing through the x– axis at
    is a horizontal line passing through the y– axis at .
  • If y is isolated on one side of the equation, graph by plotting points.
  • Choose any three values for x and then solve for the corresponding y– values.
  • If the equation is of the form , find the intercepts. Find the x– and y– intercepts and then a third point.

Glossary

Practice Makes Perfect

Identify the x– and y– Intercepts on a Graph

In the following exercises, find the x– and y– intercepts on each graph.

1.	2.
3.	4.
5.	6.
7.	8.
9.	10.
11.	12.

Find the x– and y– Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

41.	42.
43.	44.
45.	46.
47.	48.
49.	49.
51.	52.
53.	54.
55.	56.
57.	58.
59.	60.
61.	62.
63.	64.
65.	66.

Everyday Math

67. Road trip. Damien is driving from Thunder Bay to Montreal, a distance of 1000 miles. The x– axis on the graph below shows the time in hours since Damien left Thunder Bay. The y– axis represents the distance he has left to drive. a) Find the x– and y– intercepts. b) Explain what the x– and y– intercepts mean for Damien.

68. Road trip. Jenna filled up the gas tank of her truck and headed out on a road trip. The x– axis on the graph below shows the number of miles Jenna drove since filling up. The y– axis represents the number of gallons of gas in the truck’s gas tank. a) Find the x– and y– intercepts. b) Explain what the x– and y– intercepts mean for Ozzie.

Writing Exercises

69. How do you find the x– intercept of the graph of ?	70. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation ? Why?
71. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation ? Why?	72. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation ? Why?

Answers

1.	3.
5.	7.
9.	11.
13.	15.
17.	19.
21.	23.
25.	27.
29.	31.
33.	35.
37.	39.
41.	43.
45.	47.
49.	51.
53.	55.
57.	59.
61.	63.
65.	67. a) b) At , he has been gone 0 hours and has 1000 miles left. At , he has been gone 15 hours and has 0 miles left to go.
69. Answers will vary.	71. Answers will vary.

Attributions

This chapter has been adapted from “Graph with Intercepts” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?

In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

In this section, we will explore the concept of slope.

Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

Doing the Manipulative Mathematics activity “Exploring Slope” will help you develop a better understanding of the slope of a line. (Graph paper can be used instead of a geoboard, if needed.)

We’ll start by stretching a rubber band between two pegs as shown in (Figure 1).

Doesn’t it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in (Figure 2)

We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in (Figure 3).

If our geoboard and rubber band look just like the one shown in (Figure 4), the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)

The rise on this geoboard is 2, as the rubber band goes up two units.

What is the run?

The rubber band goes across 3 units. The run is 3 (see (Figure 4)).

The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter .

What is the slope of the line on the geoboard in (Figure 4)?

The line has slope . This means that the line rises 2 units for every 3 units of run.

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.

Notice that in (Example 1) the slope is positive and in (Example 2) the slope is negative. Do you notice any difference in the two lines shown in (Figure 5a) and (Figure 5b)?

Use to Find the Slope of a Line from its Graph

Now, we’ll look at some graphs on the -coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

So how do we find the slope of the horizontal line ? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this.

What is the rise?	The rise is .
Count the run.	The run is .
What is the slope?
	The slope of the horizontal line is .

All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.

The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.

Now, we’ll consider a vertical line, the line.

What is the rise?	The rise is .
Count the run.	The run is .
What is the slope?

But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line is undefined.

The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.

Use the Slope Formula to find the Slope of a Line Between Two Points

Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

We have seen that an ordered pair gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol be used to represent two different points? Mathematicians use subscripts to distinguish the points.

The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?

We will use to identify the first point and to identify the second point.

If we had more than two points, we could use , , and so on.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points and .

Since we have two points, we will use subscript notation, .

On the graph, we counted the rise of 3 and the run of 5

Notice that the rise of 3 can be found by subtracting the y-coordinates 6 and 3

And the run of 5 can be found by subtracting the x-coordinates 7 and 2

We know . So .

We rewrite the rise and run by putting in the coordinates .

But 6 is , the y-coordinate of the second point and 3 is , the y-coordinate of the first point.

So we can rewrite the slope using subscript notation.

Also, 7 is , the x-coordinate of the second point and 2 is , the x-coordinate of the first point.

So, again, we rewrite the slope using subscript notation.

We’ve shown that is really another version of . We can use this formula to find the slope of a line when we have two points on the line.

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

Key Concepts

• Find the Slope of a Line from its Graph using
  1. Locate two points on the line whose coordinates are integers.
  2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
  3. Count the rise and the run on the legs of the triangle.
  4. Take the ratio of rise to run to find the slope.
• Graph a Line Given a Point and the Slope
  1. Plot the given point.
  2. Use the slope formula to identify the rise and the run.
  3. Starting at the given point, count out the rise and run to mark the second point.
  4. Connect the points with a line.
• Slope of a Horizontal Line
  • The slope of a horizontal line, , is 0.
• Slope of a vertical line
  • The slope of a vertical line, , is undefined

Glossary

Practice Makes Perfect

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

1.	2.
3.	4.
5.	6.
7.	8.

In the following exercises, model each slope. Draw a picture to show your results.

9.	10.
11.	12.
13.	14.
15.	16.

Use to find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

17.	18.
19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

33.	34.
35.	36.
37.	38.
39.	40.

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

41.	42.
43.	44.
45.	46.
47.	48.
49.	50.
51.	52.

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

53. ;	54. ;
55. ;	56. ;
57. ;	58. ;
59. ;	60.;
61. y-intercept 3;	62. y-intercept 5;
63. x-intercept ;	64. x-intercept ;
65. ;	66. ;
67. ;	67. ;

Everyday Math

69. Slope of a roof. An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an “ 12 pitch” meaning , where is the measurement from the roof to the level—the rise. It is also sometimes stated as an “-in-12 pitch”. a) What is the slope of the roof in this picture? b) What is the pitch in construction terms?	70. The slope of the roof shown here is measured with a 12” level and a ruler. What is the slope of this roof?
71. Road grade. A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?	72. Highway grade. A local road rises 2 feet for every 50 feet of highway. a) What is the slope of the highway? b) The grade of a highway is its slope expressed as a percent. What is the grade of this highway?
73. Wheelchair ramp. The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run. a) How long must the ramp be to accommodate a 24-inch rise to the door? b) Create a model of this ramp.	74. Wheelchair ramp. A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend a ramp. a) How long must a ramp be to easily accommodate a 24-inch rise to the door? b) Create a model of this ramp.

Writing Exercises

75. What does the sign of the slope tell you about a line?	76. How does the graph of a line with slope differ from the graph of a line with slope ?
77. Why is the slope of a vertical line “undefined”?

Answers

 

1.	3.
5.	7.
9.	11.
13.	15.
17.	19.
21.	23.
25.	27.
29.	31.
33. 0	35. undefined
37. 0	39. undefined
41.	43.
45.	47.
49.	51.
53.	55.
57.	59.
61.	63.
65.	67.
69. a) b) 4 12 pitch or 4-in-12 pitch	71. ; rise = 3, run = 50
73. a) 288 inches (24 feet) b) Models will vary.	75. When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right.
77. A vertical line has 0 run and since division by 0 is undefined the slope is undefined.

Attributions

This chapter has been adapted from “Understand Slope of a Line” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables, we graphed the line of the equation by plotting points. See (Figure). Let’s find the slope of this line.

The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:

What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is . The equation of this line is:

Notice, the line has:

When a linear equation is solved for , the coefficient of the term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation is in slope–intercept form.

Identify the Slope and y-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y-intercept from the equation of a line.

When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for .

Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.

We have used a grid with and both going from about to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.

Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this chapter, and the method we used to graph each of them.

Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both and are on the same side of the equation. These two equations are of the form . We substituted to find the x-intercept and to find the y-intercept, and then found a third point by choosing another value for or .

Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.

Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.

The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labour needed to produce each item.

Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See (Figure).

Verify that both lines have the same slope, , and different y-intercepts.

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel. See (Figure).

Vertical lines with different x-intercepts are parallel.

Let’s graph the equations and on the same grid. The first equation is already in slope–intercept form: . We solve the second equation for :

Graph the lines.

Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?

The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.

Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

Use Slopes to Identify Perpendicular Lines

Let’s look at the lines whose equations are and , shown in (Figure).

These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.

What do you notice about the slopes of these two lines? As we read from left to right, the line rises, so its slope is positive. The line drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?

If we look at the slope of the first line, , and the slope of the second line, , we can see that they are negative reciprocals of each other. If we multiply them, their product is .

This is always true for perpendicular lines and leads us to this definition.

We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.

We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is , the lines are perpendicular. Perpendicular lines may have the same y-intercepts.

Key Concepts

• The slope–intercept form of an equation of a line with slope and y-intercept, is, .
• Graph a Line Using its Slope and y-Intercept
  1. Find the slope-intercept form of the equation of the line.
  2. Identify the slope and y-intercept.
  3. Plot the y-intercept.
  4. Use the slope formula to identify the rise and the run.
  5. Starting at the y-intercept, count out the rise and run to mark the second point.
  6. Connect the points with a line.
• Strategy for Choosing the Most Convenient Method to Graph a Line: Consider the form of the equation.
  • If it only has one variable, it is a vertical or horizontal line.
    is a vertical line passing through the x-axis at .
    is a horizontal line passing through the y-axis at .
  • If is isolated on one side of the equation, in the form , graph by using the slope and y-intercept.
    Identify the slope and y-intercept and then graph.
  • If the equation is of the form , find the intercepts.
    Find the x– and y-intercepts, a third point, and then graph.
• Parallel lines are lines in the same plane that do not intersect.
  • Parallel lines have the same slope and different y-intercepts.
  • If m₁ and m₂ are the slopes of two parallel lines then .
  • Parallel vertical lines have different x-intercepts.
• Perpendicular lines are lines in the same plane that form a right angle.
  • If are the slopes of two perpendicular lines, then and .
  • Vertical lines and horizontal lines are always perpendicular to each other.

Glossary

Practice Makes Perfect

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation .

1.	2.
3.	4.
5.	6.

Identify the Slope and y-Intercept From an Equation of a Line

In the following exercises, identify the slope and y-intercept of each line.

7.	8.
9.	10.
11.	12.
13.	14.
15.	16.

Graph a Line Using Its Slope and Intercept

In the following exercises, graph the line of each equation using its slope and y-intercept.

17.	18.
19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.
31.	32.

Choose the Most Convenient Method to Graph a Line

In the following exercises, determine the most convenient method to graph each line.

33.	34.
35.	36.
37.	38.
39.	40.
41.	42.
43.	44.
45.	46.
47.	48.

Graph and Interpret Applications of Slope–Intercept

49. The equation models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 0 units of water. Find the payment for a month when Randy used 15 units of water. Interpret the slope and P-intercept of the equation. Graph the equation.	50. The equation models the relation between the amount of Tuyet’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find Tuyet’s payment for a month when 0 units of water are used. Find Tuyet’s payment for a month when 12 units of water are used. Interpret the slope and P-intercept of the equation. Graph the equation.
51. Janelle is planning to rent a car while on vacation. The equation models the relation between the cost in dollars, C, per day and the number of miles, m, she drives in one day. Find the cost if Janelle drives the car 0 miles one day. Find the cost on a day when Janelle drives the car 400 miles. Interpret the slope and C–intercept of the equation. Graph the equation.	52. Bruce drives his car for his job. The equation models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 0 miles. Find the amount Bruce is reimbursed on a day when he drives 220 miles. Interpret the slope and R-intercept of the equation. Graph the equation.
53. Patel’s weekly salary includes a base pay plus commission on his sales. The equation models the relation between his weekly salary, S, in dollars and the amount of his sales, c, in dollars. Find Patel’s salary for a week when his sales were 0. Find Patel’s salary for a week when his sales were 18,540. Interpret the slope and S-intercept of the equation. Graph the equation.	54. Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars. Find Cherie’s salary for a week when her sales were 0. Find Cherie’s salary for a week when her sales were 3600. Interpret the slope and S–intercept of the equation. Graph the equation.
55. Margie is planning a dinner banquet. The equation models the relation between the cost in dollars, C of the banquet and the number of guests, g. Find the cost if the number of guests is 50. Find the cost if the number of guests is 100. Interpret the slope and C–intercept of the equation. Graph the equation.	56. Costa is planning a lunch banquet. The equation models the relation between the cost in dollars, C, of the banquet and the number of guests, g. Find the cost if the number of guests is 40. Find the cost if the number of guests is 80. Interpret the slope and C-intercept of the equation. Graph the equation.

Use Slopes to Identify Parallel Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

57.	58.
59.	60.
61.	62.
63.	64.
65.	66.
67.	68.
69.	70.
71.	72.
73.	74.
75.	76.
77.	78.
79.	80.
81.	82.

Use Slopes to Identify Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

83.	84.
85.	86.
87.	88.
89.	90.
91.	92.
93.	94.

Everyday Math

95. The equation is used to estimate the number of cricket chirps, n, in one minute based on the temperature in degrees Fahrenheit, T. Explain what the slope of the equation means. Explain what the n–intercept of the equation means. Is this a realistic situation?	96. The equation can be used to convert temperatures F, on the Fahrenheit scale to temperatures, C, on the Celsius scale. Explain what the slope of the equation means. Explain what the C–intercept of the equation means.
97. Why are all horizontal lines parallel?	98. Explain in your own words how to decide which method to use to graph a line.

Answers

2. slope and y-intercept	3. slope and y-intercept
6. slope and y-intercept	7.
10.	11.
14.	15.
18.	19.
22.	23.
26.	27.
30.	31.
34. horizontal line	35. vertical line
38. slope–intercept	39. intercepts
42. slope–intercept	43. horizontal line
46. intercepts	47. slope–intercept
50. a) $28 b) $66.10 c) The slope, 2.54, means that Randy’s payment, P, increases by $2.54 when the number of units of water he used, w, increases by 1. The P–intercept means that if the number units of water Randy used was 0, the payment would be $28. d)	51. a) $15 b) $143 c) The slope, 0.32, means that the cost, C, increases by $0.32 when the number of miles driven, m, increases by 1. The C-intercept means that if Janelle drives 0 miles one day, the cost would be $15. d)
54. a) $750 b) $2418.60 c) The slope, 0.09, means that Patel’s salary, S, increases by $0.09 for every $1 increase in his sales. The S-intercept means that when his sales are $0, his salary is $750. d)	55. a) $2850 b) $4950 c) The slope, 42, means that the cost, C, increases by $42 for when the number of guests increases by 1. The C-intercept means that when the number of guests is 0, the cost would be $750. d)
58. parallel	59. parallel
62. parallel	63. parallel
66. parallel	67. parallel
70. parallel	71. parallel
74. not parallel	75. not parallel
78. not parallel	79. not parallel
82. not parallel	83. perpendicular
86. perpendicular	87. not perpendicular
90. not perpendicular	91. perpendicular
94. perpendicular	95. a) For every increase of one degree Fahrenheit, the number of chirps increases by four. b) There would be chirps when the Fahrenheit temperature is °. (Notice that this does not make sense; this model cannot be used for all possible temperatures.)
98. Answers will vary.

1. slope and y-intercept	3. slope and y-intercept
5. slope and y-intercept	7.
9.	11.
13.	15.
17.	19.
21.	23.
25.	27.
29.	31.
33. horizontal line	35. vertical line
37. slope–intercept	39. intercepts
41. slope–intercept	43. horizontal line
45. intercepts	47. slope–intercept
49. a) $28 b) $66.10 c) The slope, 2.54, means that Randy’s payment, P, increases by $2.54 when the number of units of water he used, w, increases by 1. The P–intercept means that if the number units of water Randy used was 0, the payment would be $28. d)	51. a) $15 b) $143 c) The slope, 0.32, means that the cost, C, increases by $0.32 when the number of miles driven, m, increases by 1. The C-intercept means that if Janelle drives 0 miles one day, the cost would be $15. d)
53. a) $750 b) $2418.60 c) The slope, 0.09, means that Patel’s salary, S, increases by $0.09 for every $1 increase in his sales. The S-intercept means that when his sales are $0, his salary is $750. d)	55. a) $2850 b) $4950 c) The slope, 42, means that the cost, C, increases by $42 for when the number of guests increases by 1. The C-intercept means that when the number of guests is 0, the cost would be $750. d)
57. parallel	59. parallel
61. parallel	63. parallel
65. parallel	67. parallel
69. parallel	71. parallel
73. not parallel	75. not parallel
77. not parallel	79. not parallel
81. not parallel	83. perpendicular
85. perpendicular	87. not perpendicular
89. not perpendicular	91. perpendicular
93. perpendicular	95. a) For every increase of one degree Fahrenheit, the number of chirps increases by four. b) There would be chirps when the Fahrenheit temperature is °. (Notice that this does not make sense; this model cannot be used for all possible temperatures.)
97. Answers will vary.

Attributions

This chapter has been adapted from “Use the Slope–Intercept Form of an Equation of a Line” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.